The quantum optical description of a double Mach-Zehnder interferometer
In this paper we describe within the formalism of Quantum Optics (QO) a double Mach-Zehnder interferometer (MZI). For single photon Fock states this experimental setup is shown to exhibit a counter-intuitive behavior: for certain values of the path length difference of the first MZI, the singles photon-count statistics at the output detectors does not change, whatever the difference in path length for the second MZI. For simultaneously impinging light quanta, we show that this setup is able to show the same HOM antibunching effect previously obtained with a beam splitter. However, by adding substantial delays in each MZI, we can obtain the same effect even if the “photon wave packets” do not overlap at the second beam splitter.
The beam splitter (BS) is one of the most widely used devices in Quantum Optics. This notoriety is partially due to the fact that a beam splitter can transform a non-entangled state into an entangled one Lou03 (); Man95 (); Ger04 (); Gry10 () (and vice-versa), differentiate between a coherent and a Fock state Gra86 (); Asp91 () and reveal non-classical features of light Kim77 (); Gho87 (); HOM87 (). Applying a coherent state Gla63 (); Sud63 (); Gla63b () at one input and a single-quantum Fock state at the other one of a BS allows measuring quantum states of light using the homodyne detector Yue78 (); Yue83 (); Leo95 ().
In the classical description of a lossless beam splitter, energy conservation imposes the relation between input and output electric fields Lou03 (); Gry10 (). In the quantum optical description, fields are replaced by operators Tan97 (). A more general framework has been developed, where beam-splitters are described in SU(2) symmetries Yur86 (); Cam89 (). However, authors prefer a subset of this general model, having symmetrical Lou03 (); Ger04 () (typically for single layer dielectric beam splitters) or non-symmetrical operator input-output operator relations Man95 (); Gry10 () (typically for cube beam-splitters).
A Mach-Zehnder interferometer (MZI) is a device composed of two beam splitters and two mirrors Lou03 (). Its versatility has led to its use in countless experiments Gra86 (); Rar90 (); Fra91 (); Kwi95 (). Applying a single light quantum at one input, the rate of photo-detection oscillates as the path-length difference of the interferometer is swept Gra86 (). However, no coincidence counts are detected. Applying pairs of light quanta at its inputs, specific non-classical effects show up Rar90 ().
The special interest in Quantum Optics stems from the fact that it allowed a whole new set of Gedankenexperiments to be brought to reality (e.g. quantum eraser Scu82 (); Kim00 (); Wal02 (); Jac08 () interaction-free measurements EV93 (); Kwi95 (), quantum-non-demolition (QND) experiments Gle07 ()). A review of fundamental experiments in the field of QO is given in Steinberg et al. Ste96 ().
The so-called “semi-classical” approach Man95 (); Gry10 () in QO explains many aspects of light (including the photo-electric effect Man64 ()) however, only a full quantized theory is able to distinguish between “classical” (e.g. coherent, thermal) and “non-classical” (e.g. Fock, squeezed) states of light Gra86 (); Asp91 (). The Hanbury-Brown and Twiss Han57 () experiment that yielded the “photon bunching” effect for thermal light is expected to show anti-bunching for Fock states of light, a completely non-classical effect. The first to prove the existence of such non-classical states of light were Kimble, Dagenais and Mandel Kim77 ().
Using a beam splitter and a source of parametric down-conversion, Hong, Ou and Mandel HOM87 () experimentally proved the existence of this non-classical effect for pairs of single-quantum light states. Varying the arrival time of the “single photon wave packets” they obtained what we now call the “HOM dip”: a sharp drop in the coincidence counts when the light quanta impinge simultaneously on the beam splitter. Other variants of this experiment exist, for example with independent light sources Fea89 (); Rar05 ().
The question if the localized “photon wave packets” impinging simultaneously on the beam splitter tell the whole story arose ever since the HOM experiment. In order to answer this question, Pittman et al. Pit96 () performed the same HOM interferometer experiment but with a voluntary time delay between the “photon wave packets” at the beam splitter, compensated thereafter before the detectors. The same dip was obtained in the coincidence counting rate, thus proving that the reassuring image of overlapping “photon wave packets” at the beam splitter is not the key to this experiment and the authors conclude that “the intuitively comforting notion of the photons overlapping at the beam splitter is not at the heart of the interference, but a mere artifact of the particular geometry of the setups” Pit96 (). Kim et al. Kim99 () and later Kim Kim03 () went even further, proving that separating the “single photon pulses” beyond the coherence time of the pump laser still yields the famous HOM dip, with a visibility of more than %. The initial experiment (without delays) was also performed with orthogonal polarizations imposed to its input (i.e. and ). As expected, no interference was found, although the “photon wave packets” overlap at the beam splitter. The author concludes that “ the photon bunching picture often used in literature is indeed incorrect in general and should not be used whenever possible”.
Bylander et al. Byl03 (), using single light quanta from different sources in successive pulses show that the same phenomenon takes place, if indistinguishability is assured.
It is often believed that only identical (i.e. having the same energy) single-quantum states of light can produce this type of HOM interference. Indistinguishability Fey65 () is indeed, important, but on the detector side. Raymer et al. Ray10 () propose an interference experiment of “two photons of different color”. Using an active beam splitter, the initially distinguishable “red” and “blue” single-photon states can be converted into indistinguishable “green” single-photon states, hence the quantum interference leading to the HOM dip.
In spite of all these experiments, the localized “photon wave packet” picture is widely found today in literature, giving the impression that it is an unquestionable common place knowledge. We can read such statements as “the length of the photon wave packet”, “the photon wave packets overlap” or “the photons were short”, and we can even see graphics depicting Gaussian-damped sinusoids impinging on a beam splitter or on a detector, thus giving the impression that the very localized photon wave packet is taken more or less seriously.
We wish to add another argument in this paper in order to dispel this simplistic view of light quanta impinging on a beam splitter. However, compared to previous experiments, we propose a simple setup, requiring only photon pair production (from a parametric down-conversion source, for example), photo-detectors and regular beam splitters. No polarization beam splitters and no expensive pulse creation/selection equipment is required.
This paper is organized as follows. In Section II we describe in the formalism of Quantum Optics the beam-splitter and the Mach-Zehnder interferometer. We discuss the anti-bunching effect of a beam-splitter in Section III. The quantum-optical description of a double MZI experiment is done in Section IV. Two experiments with the double MZI experiment are proposed: in Section V the experimental setup is excited with a single-quantum Fock state while singles probability counts are evaluated and in Section VI two single-quantum states impinge simultaneously at its input while the main focus lies on the coincidence photo-counts. Finally, conclusions are drawn in Section VII.
Ii The quantum optical description of beam splitters and Mach-Zehnder interferometers
In the following we shall denote by () and, respectively, , the annihilation (creation) and, respectively, displacement operators acting at the port . The quantum state of light denotes a state with one quantum of light in mode (port) and none in mode (port) . Throughout this paper, all photo-detectors are assumed to be ideal.
ii.1 The case of monochromatic light
For a lossless beam splitter, the output annihilation operators ( and ) can be written in respect with the input field operators Lou03 () as
where and represent the transmission, and, respectively, the reflection coefficients. The input field operators ( and ) obey the usual commutation relations and where is the Kronecker delta and . We impose the same commutation relations to the output field operators and end up with the constraints
At this point we have the freedom to choose the phase of our coefficients. When dealing with a balanced (50/50) beam splitter, we shall use and Lou03 (); Ger04 (). From Eqs. (1) and (2) one can also obtain the “inverse” relations involving creation operators,
It is interesting to compare the output of a BS when at its input one applies a single-quantum Fock state and, respectively, a coherent state. In the first case we have , therefore
This is an entangled state and detecting separately photons in any of the output ports will simply yield a probability of or . However, the probability of coincident counts at its output yields
a result that is expected since we have a single quantum state that cannot yield multiple detections. In the second case, we describe the coherent state using the displacement operator relation Lou03 (); Ger04 () and we find
yielding a non-entangled state, fundamentally different from a state given by Eq. (7). Even for very small , such a state will yield a non-null coincidence rate, as proven in the experiment of Aspect and Grangier Asp91 ().
and in the case of a balanced beam splitter we have , therefore . In this case, the beam splitter transforms an entangled state into a non-entangled one, where the light quantum always leaves the beam splitter from the same port.
In the following (except when specifically stated) we will assume balanced () beam splitters.
The Mach-Zehnder interferometer (depicted in Fig. 1) is composed of two mirrors and two beam splitters. The delay models the difference in optical path lengths between the two arms of the interferometer. For monochromatic light quanta of frequency we can write where denotes the time delay introduced. We can relate it to , the path length difference of the MZI and we obviously have where is the speed of light in vacuum. The input creation operators in respect with the output ones are obtained from Eqs. (5) and (6), applied to both beam splitters. Combining them and considering the delay applied to , we end up with
A single-quantum Fock state applied to the MZI is transformed into
yielding the well-known sine-like probability of singles detection,
in respect with . A more interesting situation appears if we apply two simultaneously impinging light quanta on the first beam splitter i.e . This time, we have the output state
The probability of coincident detections at and is given by
showing that indeed, by continuously varying the coincidence probability goes back and forth between to . For we have , or, in other words, the MZI is transparent to the input state.
It is worthwhile to note that in Eq. (17) the frequency of variation of the interference fringes is twice the frequency of the input light source. This phenomenon was experimentally tested by Rarity et al. Rar90 (). They used pairs of correlated light quanta from a nm source obtained from down-converting a 413.4 nm krypton-ion laser. They found that the spatial period of the probability coincidence corresponds to the nm of the pump laser.
ii.2 The case of non-monochromatic light
Extension to a continuum of modes has been already considered Lou03 (); Blo90 (), often for two-photon states impinging on a beam splitter Fea89 (); Cam90 (). Assuming non-monochromatic but narrowband light quanta, Legero et al. Leg03 (); Leg06 () extend the result from Eq. (III.1) with a space-time domain description by considering spatio-temporal mode functions of a single-mode input radiation (input ports are labelled and ). By placing the beam-splitter at the space coordinate can be omitted. The mode functions are assumed to be normalized, so that . Then, the input electric field operators can be written as and with while the output electric fields are
The output field operators for the MZI are
If the input state is applied to our MZI, we find the probability of singles detection at given by
where this time represents the central frequency of our narrowband light quantum. We find the same oscillatory behavior from Eq. (14), however damped because of the finite bandwidth assumed for our light quantum. If we input the state to the MZI, one finds the probability of coincident counts
where corresponds to the time delay between the detections at and , related to their distance from the beam splitter . If we set in Eq. (II.2) and replace the functions and with the Gaussian expressions from Eqs. (29) and (30), after time integration one gets
where we find again the same oscillatory behavior from Eq. (17), damped however by an exponential factor.
Iii The anti-bunching effect on a beam splitter
iii.1 The case of monochromatic light
If the beam-splitter is balanced, we have , therefore the output state from Eq. (III.1) vanishes. In other words, both light quanta will always exit the beam splitter through the same port. This is the antibunching or HOM effect.
iii.2 The case of non-monochromatic light
The input state is still assumed to be . The non-monochromatic character of the light quanta will be modelled through the field operators and . The probability of joint detection at the outputs of the beam splitter at times and, respectively is
implying that indeed, for simultaneously impinging light quanta on the beam splitter, detected at the same time () , the probability of coincident counts is , regardless of the temporal shapes of and . We can take Gaussian spatio-temporal mode functions,
where quantifies the time spread of our light quanta and can be seen as the time difference of the impinging Gaussian “photon wave packets” on the beam splitter. The probability of coincident counts from Eq. (28) is easily evaluated to
and integrating it over all possible values of we get
If the photo-detectors are “slow”, we integrate over the detection time difference , yielding
This detection probability shows indeed the very famous “HOM dip”, experimentally measured by Hong, Ou and Mandel HOM87 (), where they report a time measurement of a “sub-picosecond photon wave packet”. The fact that those “photon wave packets” meet at the beam splitter seems to cause the famous dip in the rate of coincidences at the detector. Eq. (33) quantifies this result, implying that a null coincidence probability can only be achieved if i.e. if both “photon wave packets” are impinging simultaneously on the beam splitter.
Iv The quantum optical description of a double MZI experiment
In the following, we introduce and discuss a double MZI interferometer experiment (depicted in Fig. 2). The beam splitter , together with the beam splitter and the two mirrors and form the first MZI. In the lower path a delay is introduced. The second MZI is composed of the beam splitters and , together with the two corresponding mirrors and . In the lower path of this interferometer a delay is present. It is assumed that, with the corresponding delays taken out of the experiment, each MZI has equal length arms. Photo-detectors and are installed at the two outputs of .
iv.1 The case of monochromatic light
The input (creation) field operators (ignoring some common phase factors) in respect with the output field operators are (see Appendix A)
These relations will be used in the next sections in order to compute the output state of our system.
iv.2 The case of non-monochromatic light
The output field operators and expressed in respect with the input fields are (see details in Appendix B)
We obviously have and . These field operators will be used in the next sections in order to find singles and coincidence photo-detection probabilities.
V The double MZI with a single light quantum at one input
In this section we analyze the double MZI experimental setup when at its input we have a single quantum (monochromatic or non-monochromatic) Fock state.
v.1 The case of monochromatic light
The input state can be written as and taking into account Eq. (IV.1) we have
The probability of single-photon detection at the detector can be easily computed, yielding
Similarly, computing the single-photon detection probability at the detector we find
and we have , as expected. However, Eqs. (39) and (40) imply that the sine-like variation of the detection probabilities in respect with the length difference of the two arms for a single MZI (as found in Section II.1) is no longer true. Indeed, by setting for example , we end up with , no matter what value takes. The same is true for if we fix .
This apparent paradox of Eqs. (39) and (40) can be explained if we consider, for example, the state of the field at the output of beam splitter for . Indeed, using Eq. (10) we find . In other words, our light quantum always takes only one arm in the second MZI. The delay becomes therefore, useless.
If we set so that we have , the probability of single-photon detection at the detector will be , in other words, the delay modulates the photo-detection probability .
The probability of coincident counts is , an expected result since a single light quantum cannot yield multiple output detections. However if we apply a coherent source instead of the Fock state, the situation dramatically changes. Indeed, for a state we find
The rate of coincidence detection is proportional to
For the singles rates one easily finds
implying i.e. we have no antibunching with a coherent state.
v.2 The case of non-monochromatic light
If we suppose a single light quantum having some spatio-temporal extension , we characterize the probability of single-photon detection at time at the detector by
Performing the same computations for the detector yields
If we fix so that and denoting , and , we can rewrite the singles detection probabilities as
We depict in Fig. 3 and for three different values of . This parameter imposes constraints on the maximum amplitude of the sine-like behavior of the probabilities and while the Gaussian shaping causes them to “fade” towards the value of as increases. For we obviously have , whatever the value of .
Vi The double MZI with two simultaneously impinging light quanta at its inputs
In this section we describe an experiment with the double MZI depicted Fig. 2, able to show that the antibunching effect on a beam splitter has nothing to do with “photon wave packets meeting at the beam splitter”. Both the monochromatic and the non-monochromatic cases are discussed.
vi.1 The case of monochromatic light
The probability of coincident detections at the outputs detectors and is given by
If we take out both delays from the experimental setup, we end up with the same type of transformation given by Eq. (III.1) i.e. the whole experiment is equivalent to a beam splitter. At this point we can claim that the “photon wave packets” meet at the beam splitter . However, the same transformation can be obtained with the delays in place, by choosing where . These delays can be made arbitrarily large, the only limitation being imposed by the coherence properties of the light source. This way, the output state from (VI.1) becomes
i.e. a perfect anti-correlation that we expect to show the famous “HOM dip” in the case of non-monochromatic light quanta. However, this time we have no possible way of having “overlapping photon wave packets” at the beam splitter111Strictly speaking, this experiment performs exactly the opposite of Eq. (III.1), namely we have , which is, of course, perfectly equivalent to Eq. (III.1). But if we insist on finding exactly the anti-bunching effect from Eq. (III.1) at , all we have to do is add another MZI in front of the first one. .
This experiment can also be seen as a delayed “photon wave packets” HOM interferometer at the beam splitter while the two other beam splitters act as quantum erasers, so that the interfering paths at become indiscernible.
vi.2 The case of non-monochromatic light
Extending the above results to spatio-temporal modes, the probability of coincident counts at detectors and , at times and, respectively, is given by
We will be interested in the time-integrated detection probability over all and, eventually, for the non time-resolved detection, integrated over , also.
If we employ the spatio-temporal modes from Eqs. (29) and (30), after a series of rather long calculations (detailed in Appendix E), the time-integrated (over and ) probability of coincidence for yields
The surface plot of from Eq. (VI.2) versus and is depicted in Fig. 4. As expected, for , we have a dip in the probability of coincident counts. But besides this dip, there are other minima of for and . This time, we have no possible “photon wave packets” overlapping at the beam splitter .
In this paper we introduced and analyzed a new experimental setup within the standard formalism of QO. The proposed double Mach-Zehnder interferometer experiment shows a counter-intuitive behavior when excited with a single light quantum: the path length difference of one MZI “modulates” the amplitude of singles detection rates versus the path length difference of the other MZI. In the extreme case, for certain values of the path length difference of one MZI, the output singles detection rates do not change, whatever the length of the other MZI.
In the case of two simultaneously impinging light quanta at its input, the double MZI experiment is able to show the same “HOM dip” behavior for certain values of the delays and , whether or not the “photon wave packets” meet at the second beam spitter. In the extreme case of nearly monochromatic light, and may introduce arbitrarily large delays, nonetheless the “HOM dip” in the coincident counts will be present, showing that the intuitive image of “overlapping photon wave packets” at the beam splitter has a questionable physical reality, not being the key to explain this experiment.
Appendix A Computation of the input field operators in respect with the output ones
We write down the “inverse” input-output operator relations for each beam splitter. Starting with we have
For the second beam splitter we also take into account the delay , yielding
Finally, the beam splitter transforms the field operators
We group together the and terms, arriving at the final expression
Factoring out a common phase factor and replacing the complex exponentials with trigonometric functions takes us to Eq. (IV.1). Similar computations lead us to
and by the same factorization we end up with Eq. (IV.1). Please note that in Eqs. (IV.1) and (IV.1) we chose to factor out (i.e. we ended up with expressions involving sines and cosines of ). The same operations could have been done with